Editing Hausdorff maximal principle (section)

==Statement==

The Hausdorff maximal principle states that, in any [[partial order|partially ordered set]] <math>P</math>, every chain <math>C_0</math> (i.e., a [[total order|totally ordered]] [[subset]]) is contained in a maximal chain <math>C</math> (i.e., a chain that is not contained in a strictly larger chain in <math>P</math>). In general, there may be several maximal chains containing a given chain.

An equivalent form of the Hausdorff maximal principle is that in every partially ordered set, there exists a maximal chain. (Note if the set is empty, the empty subset is a maximal chain.)

This form follows from the original form since the empty set is a chain. Conversely, to deduce the original form from this form, consider the set <math>P'</math> of all chains in <math>P</math> containing a given chain <math>C_0</math> in <math>P</math>. Then <math>P'</math> is partially ordered by set inclusion. Thus, by the maximal principle in the above form, <math>P'</math> contains a maximal chain <math>C'</math>. Let <math>C</math> be the union of <math>C'</math>, which is a chain in <math>P</math> since a union of a totally ordered set of chains is a chain. Since <math>C</math> contains <math>C_0</math>, it is an element of <math>P'</math>. Also, since any chain containing <math>C</math> is contained in <math>C</math> as <math>C</math> is a union, <math>C</math> is in fact a maximal element of <math>P'</math>; i.e., a maximal chain in <math>P</math>.

The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is somehow similar to this proof. Indeed, first assume Zorn's lemma. Since a union of a totally ordered set of chains is a chain, the hypothesis of Zorn's lemma (every chain has an upper bound) is satisfied for <math>P'</math> and thus <math>P'</math> contains a maximal element or a maximal chain in <math>P</math>.

Conversely, if the maximal principle holds, then <math>P</math> contains a maximal chain <math>C</math>. By the hypothesis of Zorn's lemma, <math>C</math> has an upper bound <math>x</math> in <math>P</math>. If <math>y \ge x</math>, then <math>\widetilde{C} = C \cup \{ y \}</math> is a chain containing <math>C</math> and so by maximality, <math>\widetilde{C} = C</math>; i.e., <math>y \in C</math> and so <math>y = x</math>. <math>\square</math>